A PDE-based approach to non-dominated sorting

TL;DR

This paper introduces a PDE-based numerical scheme for non-dominated sorting in multiobjective optimization, connecting combinatorial problems to Hamilton-Jacobi equations and proposing a fast algorithm for approximate solutions.

Contribution

It presents a novel PDE approach and a fast numerical scheme for non-dominated sorting, bridging combinatorics and differential equations.

Findings

Developed a fast numerical scheme for the Hamilton-Jacobi equation

Demonstrated the scheme's effectiveness in approximate non-dominated sorting

Connected non-dominated sorting to continuum PDE models

Abstract

Non-dominated sorting is a fundamental combinatorial problem in multiobjective optimization, and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work, we showed that non-dominated sorting has a continuum limit that corresponds to solving a Hamilton-Jacobi equation. In this work we present and analyze a fast numerical scheme for this Hamilton-Jacobi equation, and show how it can be used to design a fast algorithm for approximate non-dominated sorting.